Strong illposedness for SQG in critical Sobolev spaces

We prove that the inviscid surface quasi-geostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^2({\text{bbT}}^2)$ without any solutions in $L_t^{\mathrm{\infty }}{H}^2$. Moreover, we prove strong critical norm inflation for $C^{\mathrm{\infty }}$–smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with two-dimensional incompressible Euler equations.